Finite Group Databasebeta

← C7×#24122 #13201338

Subgroups — C7×#24122 #13201338

Order 1792.

1134 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic
2C2×C7×#477C211yesmaximal, characteristic, complemented
3C2×C4×C7×Q16C211yesmaximal, characteristic, complemented
4C7×#2538C211yesmaximal, characteristic
5C7×#2036C212yesmaximal, complemented
6C7×#2001C212yesmaximal, complemented
7C7×#1990C211yesmaximal, characteristic, complemented
8C7×#1954C211yesmaximal, characteristic
9C7×#1872C212yesmaximal, complemented
10C7×#1852C212yesmaximal, complemented
11C7×#1803C211yesmaximal, characteristic, complemented
12C22×C7×Q16C2211yescharacteristic
13C7×#554—24no—
14C7×#554—24no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125—21no—
17C2×C7×C22⋊Q8—22no—
18C2×C7×C22⋊Q8C2212yes—
19C2×C7×C22⋊Q8C2212yes—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×Q8C2211yescharacteristic
22C2×C4×C7×D4—21no—
23C2×C4×C7×D4C2211yescharacteristic
24C22×C7×C4⋊C4C2212yes—
25C7×#477—24no—
26C4×C7×Q16—24no—
27C2×C7×#107C2211yescharacteristic
28C2×C7×#107C2212yes—
29C2×C7×C4⋊C8C2211yescharacteristic
30C2×C7×#103C2212yes—
31C2×C7×#103C2211yescharacteristic
32C2×C7×#103C2211yescharacteristic
33C2×C7×#103C2212yes—
34C2×C7×C22⋊C8C2212yes—
35C2×C7×C22⋊C8C2212yes—
36C2×C4×C7×C8C2211yescharacteristic
37C7×#396—22no—
38C7×#390—21no—
39C7×#389—21no—
40C7×#384C2211yescharacteristic
41C7×#380C2212yes—
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45#24122C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8C2311yescharacteristic
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4D411yescharacteristic
50C23×C4×C7—22no—
51C23×C4×C7C2312yes—
52C2×C7×Q16—22no—
53C2×C7×Q16—22no—
54C2×C7×Q16—22no—
55C22×C7×C8C2311yescharacteristic
56C22×C7×C8C2312yes—
57C22×C7×C8C2311yescharacteristic
58C7×C4⋊Q8—22no—
59C7×C4⋊Q8D414yes—
60C7×#125—28no—
61C7×C22⋊Q8—216no—
62C7×C22⋊Q8—24no—
63C7×C22⋊Q8—24no—
64C7×C22⋊Q8—24no—
65C7×C22⋊Q8—42no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C4×C7×Q8—22no—
69C4×C7×Q8D414yes—
70C4×C7×D4—28no—
71C4×C7×D4—22no—
72C4×C7×D4—22no—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—24no—
78C2×C7×C4⋊C4—24no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4D411yescharacteristic
84C2×C7×C4⋊C4C2312yes—
85C2×C7×C4⋊C4C2311yescharacteristic
86C2×C7×C4⋊C4C2311yescharacteristic
87C2×C7×C4⋊C4C2312yes—
88C2×C7×C22⋊C4—22no—
89C2×C7×C22⋊C4—22no—
90C2×C7×C22⋊C4—22no—
91C2×C7×C22⋊C4D412yes—
92C2×C42×C7—21no—
93C2×C42×C7—21no—
94C2×C42×C7C2311yescharacteristic
95C7×#107—24no—
96C7×#107—22no—
97C7×C4⋊C8—22no—
98C7×#103—22no—
99C7×#103—24no—
100C7×#103—22no—
101C7×#103—42no—
102C7×C22⋊C8—24no—
103C7×C22⋊C8—42no—
104C4×C7×C8—22no—
105C7×#95—22no—
106C2×#554C2×C711yescharacteristic
107C2×#477C2×C711yescharacteristic, complemented
108C2×C4×Q16C2×C711yescharacteristic, complemented
109#2538C2×C711yescharacteristic
110#2036C2×C712yescomplemented
111#2001C2×C712yescomplemented
112#1990C2×C711yescharacteristic, complemented
113#1954C2×C711yescharacteristic
114#1872C2×C712yescomplemented
115#1852C2×C712yescomplemented
116#1803C2×C711yescharacteristic, complemented
117C24×C7C2×D412yes—
118C2×C7×Q8—24no—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8—21no—
129C2×C7×Q8—81no—
130C2×C7×Q8—81no—
131C2×C7×Q8C2×D412yes—
132C2×C7×Q8#4112yes—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4Q1614yescomplemented
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—22no—
140C22×C4×C7—28no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—21no—
148C22×C4×C7—22no—
149C22×C4×C7—82no—
150C22×C4×C7—82no—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C7×Q16—28no—
156C2×C7×C8—21no—
157C2×C7×C8—21no—
158C2×C7×C8—21no—
159C2×C7×C8—22no—
160C2×C7×C8—22no—
161C2×C7×C8—21no—
162C2×C7×C8—22no—
163C2×C7×C8—21no—
164C2×C7×C8#4112yes—
165C7×C4⋊C4—28no—
166C7×C4⋊C4—28no—
167C7×C4⋊C4—24no—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—44no—
177C7×C4⋊C4—42no—
178C7×C4⋊C4—42no—
179C7×C4⋊C4—81no—
180C7×C4⋊C4C2×D412yes—
181C7×C4⋊C4C2×D414yes—
182C7×C22⋊C4—28no—
183C7×C22⋊C4—24no—
184C7×C22⋊C4—44no—
185C7×C22⋊C4—82no—
186C42×C7—24no—
187C42×C7—42no—
188C42×C7C2×D412yes—
189C42×C7C2×D412yes—
190C22×Q16C22×C711yescharacteristic
191#554—24no—
192#554—24no—
193C2×C4⋊Q8C22×C711yescharacteristic
194C2×#125—21no—
195C2×C22⋊Q8—22no—
196C2×C22⋊Q8C22×C712yes—
197C2×C22⋊Q8C22×C712yes—
198C2×C4×Q8C22×C711yescharacteristic
199C2×C4×Q8C22×C711yescharacteristic
200C2×C4×D4—21no—
201C2×C4×D4C22×C711yescharacteristic
202C22×C4⋊C4C22×C712yes—
203#477—24no—
204C4×Q16—24no—
205C2×#107C22×C712yes—
206C2×#107C22×C711yescharacteristic
207C2×C4⋊C8C22×C711yescharacteristic
208C2×#103C22×C712yes—
209C2×#103C22×C711yescharacteristic
210C2×#103C22×C711yescharacteristic
211C2×#103C22×C712yes—
212C2×C22⋊C8C22×C712yes—
213C2×C22⋊C8C22×C712yes—
214C2×C4×C8C22×C711yescharacteristic
215#396—22no—
216#390—21no—
217#389—21no—
218#384C22×C711yescharacteristic
219#380C22×C712yes—
220#340C22×C712yes—
221#340C22×C712yes—
222#340C22×C712yes—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7C2×Q1614yes—
230C23×C7C22×D411yescharacteristic
231C7×Q8—22no—
232C7×Q8—22no—
233C7×Q8—22no—
234C7×Q8—22no—
235C7×Q8—41no—
236C7×Q8—41no—
237C7×Q8—41no—
238C7×Q8—41no—
239C7×Q8—84no—
240C7×D4—28no—
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—24no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—41no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—41no—
255C2×C4×C7—42no—
256C2×C4×C7—41no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—41no—
266C2×C4×C7—42no—
267C2×C4×C7—41no—
268C2×C4×C7—88no—
269C2×C4×C7#12012yes—
270C2×C4×C7#12012yes—
271C2×C4×C7C2×Q1612yes—
272C2×C4×C7#13712yes—
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7C2×#4111yescharacteristic
277C2×C4×C7H(5,𝔽2)11yescharacteristic
278C2×C4×C72−1+411yescharacteristic
279C7×C8—22no—
280C7×C8—41no—
281C7×C8—42no—
282C22×Q8—21no—
283C22×Q8C23×C711yescharacteristic
284C22×Q8C23×C711yescharacteristic
285C22×D4C7×D411yescharacteristic
286C23×C4—22no—
287C23×C4C23×C712yes—
288C2×Q16—22no—
289C2×Q16—22no—
290C2×Q16—22no—
291C22×C8C23×C711yescharacteristic
292C22×C8C23×C712yes—
293C22×C8C23×C711yescharacteristic
294C4⋊Q8—22no—
295C4⋊Q8C7×D414yes—
296#125—28no—
297C22⋊Q8—216no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—24no—
301C22⋊Q8—42no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—22no—
305C4×Q8C7×D414yes—
306C4×D4—28no—
307C4×D4—22no—
308C4×D4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4C7×D411yescharacteristic
320C2×C4⋊C4C23×C711yescharacteristic
321C2×C4⋊C4C23×C712yes—
322C2×C4⋊C4C23×C711yescharacteristic
323C2×C4⋊C4C23×C712yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4C7×D412yes—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42C23×C711yescharacteristic
331#107—22no—
332#107—24no—
333C4⋊C8—22no—
334#103—22no—
335#103—22no—
336#103—24no—
337#103—42no—
338C22⋊C8—24no—
339C22⋊C8—42no—
340C4×C8—22no—
341#95—22no—
342C22×C7—24no—
343C22×C7—24no—
344C22×C7—24no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7C2×#12011yescharacteristic
350C22×C7#54611yescharacteristic
351C22×C7#54711yescharacteristic
352C22×C7C22×Q1611yescharacteristic
353C22×C7C2×#13711yescharacteristic
354C22×C7#57611yescharacteristic
355C22×C7#57711yescharacteristic
356C4×C7—22no—
357C4×C7—42no—
358C4×C7—42no—
359C4×C7—42no—
360C4×C7—42no—
361C4×C7—44no—
362C4×C7—82no—
363C4×C7—81no—
364C4×C7#45114yes—
365C4×C7#52511yescharacteristic
366C4×C7#53711yescharacteristic
367C4×C7#54811yescharacteristic
368C4×C7#55011yescharacteristic
369C24C2×C7×D412yes—
370C2×Q8—24no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—81no—
382C2×Q8—81no—
383C2×Q8C2×C7×D412yes—
384C2×Q8C7×#4112yes—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4C7×Q1614yescomplemented
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—28no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4C2×C7×D412yes—
404C22×C4C2×C7×D411yescharacteristic
405C22×C4C2×C7×D411yescharacteristic
406C22×C4C24×C711yescharacteristic
407Q16—28no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—21no—
414C2×C8—22no—
415C2×C8—21no—
416C2×C8C7×#4112yes—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—81no—
432C4⋊C4C2×C7×D412yes—
433C4⋊C4C2×C7×D414yes—
434C22⋊C4—28no—
435C22⋊C4—24no—
436C22⋊C4—44no—
437C22⋊C4—82no—
438C42—24no—
439C42—42no—
440C42C2×C7×D412yes—
441C42C2×C7×D412yes—
442C2×C7—44no—
443C2×C7#240711yescharacteristic
444C2×C7C2×#45111yescharacteristic
445C2×C7#301211yescharacteristic
446C2×C7#329911yescharacteristic
447C2×C7#333311yescharacteristic
448C2×C7#333811yescharacteristic
449C2×C7#335611yescharacteristic
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23C2×C7×Q1614yes—
457C23C22×C7×D411yescharacteristic
458Q8—22no—
459Q8—22no—
460Q8—22no—
461Q8—22no—
462Q8—41no—
463Q8—41no—
464Q8—41no—
465Q8—41no—
466Q8—84no—
467D4—28no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—88no—
496C2×C4C7×#12012yes—
497C2×C4C7×#12012yes—
498C2×C4C2×C7×Q1612yes—
499C2×C4C7×#13712yes—
500C2×C4C22×C7×D411yescharacteristic
501C2×C4C22×C7×D411yescharacteristic
502C2×C4C2×C7×#4111yescharacteristic
503C2×C4C2×C7×#4111yescharacteristic
504C2×C4C7×H(5,𝔽2)11yescharacteristic
505C2×C4C7×2−1+411yescharacteristic
506C8—22no—
507C8—41no—
508C8—42no—
509C7#2412211yescharacteristic, direct product factor, Sylow
510C22—24no—
511C22—24no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22C2×C7×#12011yescharacteristic
518C22C7×#54611yescharacteristic
519C22C7×#54711yescharacteristic
520C22C22×C7×Q1611yescharacteristic
521C22C2×C7×#13711yescharacteristic
522C22C7×#57611yescharacteristic
523C22C7×#57711yescharacteristic
524C4—22no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—44no—
530C4—81no—
531C4—82no—
532C4C7×#45114yes—
533C4C7×#52511yescharacteristic
534C4C7×#53711yescharacteristic
535C4C7×#54811yescharacteristic
536C4C7×#55011yescharacteristic
537C2—44no—
538C2C7×#240711yescharacteristic
539C2C2×C7×#45111yescharacteristic
540C2C7×#301211yescharacteristic
541C2C7×#329911yescharacteristic
542C2C7×#333311yescharacteristic
543C2C7×#333811yescharacteristic
544C2C7×#335611yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C7×#477C2semidirect
3. C2×C4×C7×Q16C2semidirect
5. C7×#2036C2semidirect
6. C7×#2001C2semidirect
7. C7×#1990C2semidirect
9. C7×#1872C2semidirect
10. C7×#1852C2semidirect
11. C7×#1803C2semidirect
45. #24122C7direct (N × H)
107. C2×#477C2×C7semidirect
108. C2×C4×Q16C2×C7semidirect
110. #2036C2×C7semidirect
111. #2001C2×C7semidirect
112. #1990C2×C7semidirect
114. #1872C2×C7semidirect
115. #1852C2×C7semidirect
116. #1803C2×C7semidirect
135. C2×C7×D4Q16semidirect
387. C2×D4C7×Q16semidirect
509. C7#24122direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C7×#5542
C2×C7×#4772
C2×C4×C7×Q162
C7×#25382
C7×#20362
C7×#20012
C7×#19902
C7×#19542
C7×#18722
C7×#18522
C7×#18032
#241227